I do not really understand what $\mathbb{Q}(\pi)$ is here:
Ofcourse we see that $\mathbb{Q}(\pi)$ is a field. But I have to "guesses" of what they mean, is one of them correct?
1. $\mathbb{Q}(\pi)$ consists of $a_n\pi^n \ldots +a_1\pi+a_0, a_i \in \mathbb{Q}$
2.$\mathbb{Q}(\pi)$ consists of $\frac{a_n\pi^n...+a_0}{b_m\pi^n +a_0}$.Where the a's and b's are rational.
In either way $\mathbb{Q}(x)$ is a set of real numbers, that can be written as a comnation of rational numbers and $\pi$?
The reason I am unclear, is that I don't really see how they use case 2 in the example.
Your second guess is a correct description of $\Bbb Q(\pi)$ (as long as you specify that the denominator is not the zero polynomial). The first guess describes the ring $\Bbb Q[\pi]$, which does not have inverses. In particular, $\pi$ has no inverse since $\pi f(\pi) \ne 0$ for any polynomial $f\in \Bbb Q[x]$. It's not hard to see that these rational functions in $\pi$ form the smallest subfield of $\Bbb C$ (or $\Bbb R$) which contains $\pi$ and $\Bbb Q.
Here, the key is that $\Bbb Q(\pi)$ is isomorphic to $\Bbb Q(x)$ as fields, they're not the same thing per se. The application of Case 2 is that $\Bbb Q(\pi)$ is the field of fractions of $\Bbb Q[\pi]$, and so in this case that evaluation map $\Bbb Q(x) \to \Bbb Q(\pi)$ is an isomorphism.